Math July 13, 2026 · 8 Min Read

Catenary Curve Calculator – Guide & Formulas

Calculate catenary curve parameters for hanging cables and chains. Enter the span, sag, and cable length to see the curve equation, coordinates, and physical properties.

Calculate catenary curve parameters with our free online calculator. Enter span and sag to see the curve equation, coordinates, cable length, and physical properties.

Key Takeaway

Use the free Catenary Curve Calculator to calculate catenary curve parameters for hanging cables and chains. enter the span, sag, and cable length to see the curve equation, coordinates, and physical properties. Get instant results with step-by-step explanations.

How to Use the Catenary Curve Calculator

  1. Step 1: Enter the horizontal span (distance between supports)
  2. Step 2: Enter the sag (vertical drop from support to lowest point)
  3. Step 3: Click Calculate to see the catenary parameter a and curve equation
  4. Step 4: Review the curve coordinates, cable length, and tension properties

The Formula

Catenary equation: y = a × cosh(x/a), where a is the catenary parameter. For a cable of span S and sag d: a = S²/(8d) + d/2. The cable length L = 2a × sinh(S/(2a)).

Variable Definitions

  • a: The catenary parameter — determines the curve shape
  • cosh: Hyperbolic cosine function: cosh(x) = (eˣ + e⁻ˣ)/2
  • sinh: Hyperbolic sine function: sinh(x) = (eˣ - e⁻ˣ)/2
  • S: The horizontal span between the two supports
  • d: The sag — vertical distance from support level to lowest point
  • L: The total cable length along the curve

Catenary for a 100m Span with 10m Sag

Find the catenary parameter, cable length, and curve coordinates.

  1. Step 1: Identify inputs. Span S = 100m, Sag d = 10m.
  2. Step 2: Calculate parameter a: a = 100²/(8×10) + 10/2 = 10000/80 + 5 = 125 + 5 = 130.
  3. Step 3: Catenary equation: y = 130 × cosh(x/130).
  4. Step 4: Cable length: L = 2 × 130 × sinh(100/(2×130)) = 260 × sinh(0.3846) ≈ 260 × 0.3939 ≈ 102.4m.
  5. Step 5: The cable is approximately 102.4m long for a 100m span with 10m sag.

Frequently Asked Questions

What is a catenary curve?

A catenary is the shape formed by a flexible chain or cable hanging freely under its own weight between two supports. It is described by the hyperbolic cosine function, not a parabola.

How is a catenary different from a parabola?

While they look similar, a catenary is y = a×cosh(x/a) and a parabola is y = ax². A catenary is the natural shape of a hanging chain, while a parabola approximates it when the sag is small relative to the span.

What does the catenary parameter a represent?

The parameter a determines the curve shape. A larger a means a shallower curve (less sag relative to span). It equals the y-coordinate at x = 0 (the lowest point of the curve).

What real-world structures follow a catenary curve?

Hanging cables, power lines, anchor chains, suspension bridge main cables (when unloaded), and catenary arches in architecture all follow catenary curves.

How do I find the cable length of a catenary?

The cable length L = 2a × sinh(S/(2a)), where a is the catenary parameter and S is the horizontal span. This accounts for the curved path, so it is always longer than the span.

Can I use this for power line sag calculations?

Yes. Power line engineers use catenary equations to calculate conductor sag, tension, and clearance. The cable weight and tension determine the catenary parameter a.